Non-residues and primitive roots in beatty sequences

William D. Banks*, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)


We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence Bα,β = {⌊αn + β⌋ : n = 1, 2, 3, . . .}, where α, β ∈ ℝ, and α is irrational. In particular, our bounds imply that for every fixed ε > 0, if p is sufficiently large and p1/2+ε ≤ N ≤ p, then among the first N elements of Bα,β, there are N/2+o(N) quadratic non-residues modulo p. When more information is available about the Diophantine properties of α, then the error term o(N) admits a sharper estimate.

Original languageEnglish
Pages (from-to)433-443
Number of pages11
JournalBulletin of the Australian Mathematical Society
Issue number3
Publication statusPublished - Jun 2006


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